Singularities in General Relativity

该研究项目研究爱因斯坦真空方程解中的时空奇点，这是一个描述无物质位置引力的非线性偏微分方程系统。这个方程组的解可以从规则的初始数据中发展出奇点;一些这样的奇点在黑洞的事件视界内。一个基本的数学问题涉及理解这样的奇点是如何从一般的初始数据中产生的。特别是，该项目探讨的问题包括：黑洞内部的边界是什么样子的？他们总是单一的吗？是否存在不在黑洞区域的奇点？这些都是广义相对论中的强和弱宇宙审查理论试图解决的问题--对这些理论的研究是这个项目的中心动机。这个项目的一个特别重点是理解黑洞内部的奇异边界，以及理解被困表面（与黑洞密切相关的结构）可能形成的尖锐标准。研究者和合作者最近的工作为球对称的爱因斯坦-麦克斯韦（真实的）标量场的强宇宙审查猜想提供了解决方案，表明从一般数据产生的解决方案具有奇异的黑洞内部，在这个意义上，时空度量不能以适当的规则方式扩展。这个项目的目的是将这些结果扩展到没有对称性假设的情况，特别是给出一个完整的描述所产生的显式克尔解的扰动黑洞的边界。在另一个方向上，该项目还旨在建立保证捕获表面形成的标准，作为理解广义相对论中弱宇宙审查猜想的一步。

This research project studies spacetime singularities in the solutions to the Einstein vacuum equations, a system of nonlinear partial differential equations that describes gravity in locations devoid of matter. Solutions to this system of equations can develop singularities from regular initial data; some such singularities are inside a black hole event horizon. A fundamental mathematical question concerns understanding how such singularities arise from generic initial data. In particular, this project explores questions that include: What does the boundary of the interior of black holes look like? Are they always singular? Are there singularities that are not in the black hole regions? These are questions that the strong and weak cosmic censorship conjectures in general relativity seek to address -- study of those conjectures is the central motivation for this project.A particular focus of the project is to understand the singular boundary of the interior of black holes and also to understand sharp criteria under which trapped surfaces (structures closely related to black holes) may form. Recent work of the investigator and collaborators provides a resolution of the strong cosmic censorship conjecture for the Einstein-Maxwell (real) scalar field in spherical symmetry, showing that solutions arising from generic data have singular black hole interiors, in the sense that the spacetime metric cannot be extended in a suitably regular manner. This project aims to extend these results to settings without symmetry assumptions, in particular to give a complete description of the boundary of black holes that arise from perturbations of the explicit Kerr solutions. In a different direction, this project also aims to establish criteria that guarantee that trapped surfaces form, as a step towards understanding the weak cosmic censorship conjecture in general relativity.